Taking the Fear
out of Math
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#1
Introduction
to the
Adjective/Noun
Theme.
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The basis of “Math as a Second Language”
is that most students see numbers as
quantities.
If you ask students
to tell you what the
number 3 is, they
might hold up
3 fingers.
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In other words, we have seen 3 fingers,
3 apples, 3 tally marks, etc. but never
“threeness” by itself.
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Definition
A quantity is a phrase consisting of
an adjective and a noun.
The adjective is a number, and
the noun is the unit.
3 fingers is a quantity in which the
adjective is 3 and the noun (unit) is fingers.
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Definition
In a similar way, 3 inches is a quantity
in which the adjective is 3, and
the noun (unit) is inches.
Key Point
As quantities, 3 fingers is not the same as
3 inches. However, as adjectives, the “3”
in “3 fingers” means the same thing as the
“3” in “3 inches”.
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Definition
The above concept transcends
mathematics. Although a blue pencil
doesn’t look like a blue sweater,
the adjective “blue” means
the same thing in each case.
Hence, at least in English grammar,
it is rather vague for someone to say
“This is a blue”.
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With the above concept in mind, our
innovative approach to teaching basic
mathematics, which we call
“Mathematics as a Second Language”,
is to introduce numbers in the same way
that people from all walks of life use them;
namely as adjectives that modify nouns.
Our technique is to show that by using this
concept, all of basic arithmetic can be
done by just knowing the addition and
multiplication tables from 0 through 9.
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The greatest obstacle to this approach
is the tendency for presenting numbers to
students only in the form of adjectives.
That is, we often will talk about 3 without
reference to what noun 3 is modifying.1
Since the noun is usually omitted, we have
to understand a few things about quantities.
note
1 In our opinion it is amazing how much clearer the various computations in both
arithmetic and algebra become when students are allowed to visualize the adjectives
as modifying nouns of their own choosing.
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First…
When we write such apparently simple
statements as 1 = 1, we are assuming
that the 1 on one side of the equal sign
is modifying the same noun as the 1 on
the other side of the equal sign.
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Secondly…
1 inch ≠ 1 mile2, even though as
an adjective the 1 that modifies “inch”
means the same thing as the 1 that
modifies “mile”.
note
2 To negate a relationship, it is a common mathematical procedure to put a “slash
mark” through the symbol that expresses the relationship. Thus, to negate a
statement such as b = c, we would write b ≠ c, which we read as “b is not equal to c”
or “b is unequal to c”.
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Thirdly…
On the other hand, as adjectives 12 ≠ 1,
but it is true that 12 inches = 1 foot.
There are other interesting things that occur
when we study the arithmetic of quantities
that we will mention briefly here but explore
in greater detail as the course unfolds.
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When we write that 3 + 2 = 5, we are
assuming that 3, 2, and 5 are modifying
the same noun.
3 dimes + 2 nickels = 40 cents, but as
adjectives it is false that 3 + 2 = 40.3
note
3 Of course if the nouns are present, it is possible that 3 + 2 = 5 even if the nouns
aren’t all the same. For example, 3 dimes + 2 nickels = 5 coins. However, if we are
thinking in terms of the amount of money, 5 coins doesn’t mean the same things as
3 dimes and 2 nickels. On the other hand, if we are thinking in terms of the number
of coins it does make sense to replace “dimes” and “nickels” by “coins” and write
3 coins + 2 coins = 5 coins.
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Why it is Important!
The fact that 3 + 2 = 5 whenever 3, 2, and 5
modify the same noun is extremely
important because it can be used to
explain many things in a simple manner.
For example, young students might be
overwhelmed by an addition problem
such as 3,000,000,000 + 2,000,000,000
because of the number of digits.
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However, this problem is simply the
place value version of 3 billion + 2 billion
for which the answer is 5 billion because
the 3, 2, and 5 are each modifying “billion”.
Based on how we add quantities, one does
not have to know what a gloog is to know
that…
3 gloogs + 2 gloogs = 5 gloogs.
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Something similar to this occurs in a
beginning algebra course when students
are asked to simplify 3x + 2x. We do not
have to know what number x represents in
order to know that 3 of them plus 2 more of
them is 5 of them.
In demonstrating that…
3 dimes + 2 nickels = 40 cents,
we changed dimes and nickels to a
common denomination (cents).
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The same thing happens when
we add fractions.
For example, to add 3/7 and 2/5, think
of the problem as being written in the
form 3 sevenths + 2 fifths.
We cannot add the 3 and the 2 because
they are modifying different units
(sevenths and fifths).
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Report Card
On a report card if you
got 3 A’s and 2 B’s you
do not say that you got
5 AB’s. You simply
say that you got 3 A’s
and 2 B’s.4
Grading Period
1
Reading
B
Language
B
Mathematics
A
Science
A
Social Studies
A
2
3
4
note
4 Schools have solved the problem of adding A’s and B’s by going to a 4.0 grade
point scale. An A is worth 4 points and a B is worth 3 points. Without going into how
the computation is formed, the student with 3 A’s and 2 B’s gets a GPA
(grade point average) of 3.6.
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The statement 3 × 2 = 6 is always true,
but what the 6 modifies depends on what
the 3 and the 2 are modifying.
3 feet × 2 pounds = 6 foot pounds 5
3 kilowatts × 2 hours = 6 kilowatt hours
3 hundred × 2 thousand = 6 hundred thousand
note
5 When we multiply 2 quantities, we multiply the two adjectives (numbers) to obtain
the adjective part of the product, and we multiply the 2 nouns (which we do my
writing them side by side) to obtain the noun part of the product.
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Why it is Important!
In doing multiplication problems of the
form…
300 × 2,000
…students mechanically multiply the 3 by
the 2 to obtain 6 and then annex the total
number of 0’s to obtain 600,000.
However, as seen above, our
adjective/noun theme gives us the answer
in an easy to understand format.
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In multiplying two fractions, we
multiply the two numerators to obtain the
numerator of the product, and we multiply
the two denominators to obtain
the denominator of the product.
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In terms of our adjective/noun theme,
the reason is that the numerators are
the adjectives and the denominators
are the nouns.6
note
6 The rule for multiplying two fractions might seem “self evident”. However, the
“rule” doesn’t work when we add two fractions. Namely, we can only add the
numerators (i.e., the adjectives) if they modify the same noun (i.e., denominator).
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In algebra, if we are given a problem
such as 3x + 2y, students often want to add
the 3 and 2, not recognizing that the 3 is
modifying x and the 2 is modifying y.
However, using our above “rule”, when we
multiply 3x by 2y, we multiply 3 by 2 to
obtain 6 and we multiply x and y
(which we may view as the nouns) by
writing them side by side.
In other words…
3x + 2y ≠ 5xy, but 3x × 2y = 6xy.
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A rather nice way to have students
see the difference between adding and
multiplying is to have them compare how
we add 3 tens and 2 tens with how we
multiply 3 tens by 2 tens.
It follows rather simply that…
3 tens + 2 tens = 5 tens.
However, 3 tens × 2 tens ≠ 6 tens.
Rather, 3 tens × 2 tens = 6 “ten tens”.
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According to our rule, (multiply the
adjectives and multiply the nouns)…
3 tens × 2 tens = 6 “ten tens”.
And since ten tens is equal to a hundred we
see that…
3 tens × 2 tens = 6 hundreds.7
note
7 Don’t confuse 3 tens × 2 tens with 3 × 2 tens.
If we take 2 tens, 3 times (that is 3 × 2 tens) the answer is 6 tens.
However, 3 tens × 2 tens = 30 × 20 = 600 = 6 hundred.
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